Approaches to Quantum Gravity

240 T. Thiemann
spectrum of all the Ĉ I , there is no non-trivial solution ∈ H Kin to the system of
quantum constraint equations Ĉ I = 0 for all I which is the quantum analogue of
the classical system of constraint equations C I = 0 for all I (because this would
mean that is a common zero eigenvector). For instance, the operator id/dx on
L 2 (R, dx) has spectrum R but none of the formal “eigenvectors” exp(−ikx) with
eigenvalue k is normalizable. Thus, the solution to the constraints has to be understood
differently, namely in a generalised sense. This comes at the price that the solutions
must be given a new Hilbert space inner product with respect to which they are
normalisable.
We will now present a method to solve all the constraints and to construct an inner
product induced from that of H Kin in a single stroke, see [12] and [13] formore
details. Consider the Master constraint
M := ∑ IJ
C I K IJ C J (13.5)
where K IJ is a positive definite matrix which may depend non-trivially on the phase
space and which decays sufficiently fast so that M is globally defined and differentiable
on M. It is called the Master constraint because obviously M = 0 ⇔ C I =
0 ∀I . The concrete choice of K IJ is further guided by possible symmetry properties
that M is supposed to have and by the requirement that the corresponding Master
constraint operator ̂M is densely defined on H Kin . As a first check, consider the case
that the point zero is only contained in the point spectrum of every Ĉ I and define
̂M := ∑ I K I Ĉ † I ĈI where K I > 0 are positive numbers. Obviously, Ĉ I = 0 for all
I implies ̂M = 0. Conversely, if ̂M = 0 then 0 =< ,̂M >= ∑ I K I ||Ĉ I || 2
implies Ĉ I = 0 for all I . Hence, in the simplest case, the single Master constraint
contains the same information as the system of all constraints.
Let us now consider the general case and assume that ̂M has been quantised as a
positive self-adjoint operator on H Kin . 4 Then it is a well known fact that the Hilbert
space H Kin is unitarily equivalent to a direct integral of Hilbert spaces subordinate to
̂M, that is,
H Kin
∼ =
∫ ⊕
R + dμ(λ) H ⊕ (λ) =: H ⊕ μ,N . (13.6)
Here the Hilbert spaces H ⊕ (λ) are induced from H Kin and by the choice of the
measure μ and come with their own inner product. One can show that the measure
class [μ] and the function class [N], where N(λ) = dim(H ⊕ (λ)) is the multiplicity
of the “eigenvalue” λ, are unique 5 and in turn determine ̂M uniquely up to unitary
4 Notice that ̂M is naturally quantised as a positive operator and that every positive operator has a natural selfadjoint
extension, the so-called Friedrichs extension [14].
5 Two measures are equivalent if they have the same measure zero sets. Two measurable functions are equivalent
if they agree up to measure zero sets.